Compact Sobolev imbeddings for unbounded domains

Extrapolation of Reduced Sobolev Imbeddings

We consider fractional Sobolev spaces with dominating mixed derivatives and prove generalizations of Trudinger’s limiting imbedding theorem.

On the constants for some Sobolev imbeddings

We consider the imbedding inequality ‖ ‖ Lr(R d ) ≤ Sr,n,d ‖ ‖Hn(Rd); H n(R) is the Sobolev space (or Bessel potential space) of L2 type and (integer or fractional) order n. We write down upper bounds for the constants Sr,n,d, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r = 2 ...

Criteria for Imbeddings of Sobolev-poincaré Type

Ω u dx and Ω ⊂ R is bounded and satisfies a uniform interior cone condition; by density of smooth functions (1.1) then holds for all functions in the Sobolev space W (Ω) consisting of all functions in L(Ω) whose distributional gradients belong to L(Ω). For 1 < p < n, inequality (1.1) was proved by Sobolev ([So1], [So2]); for p = 1, it is due to Gagliardo [G] and Nirenberg [N] (also see [M1]). W...

On weighted critical imbeddings of Sobolev spaces

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that Lexp tα(χB) = Lexp tα(ρ) if and only if ρq ∈ Lq with some q > 1. Second, we consider the Sobolev space W ...

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains